Research Papers

Numerical Approximation of Tangent Moduli for Finite Element Implementations of Nonlinear Hyperelastic Material Models

[+] Author and Article Information
Wei Sun1

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269weisun@engr.uconn.edu

Elliot L. Chaikof

Department of Surgery, and Department of Biomedical Engineering, Emory University and Georgia Institute of Technology, Atlanta, GA

Marc E. Levenston

Department of Mechanical Engineering, Stanford University, Stanford, CA


Corresponding author. Also at Biomedical Engineering, 203 Bronwell Building, University of Connecticut, Storrs, CT 06269-3139.

J Biomech Eng 130(6), 061003 (Oct 09, 2008) (7 pages) doi:10.1115/1.2979872 History: Received October 30, 2007; Revised May 26, 2008; Published October 09, 2008

Finite element (FE) implementations of nearly incompressible material models often employ decoupled numerical treatments of the dilatational and deviatoric parts of the deformation gradient. This treatment allows the dilatational stiffness to be handled separately to alleviate ill conditioning of the tangent stiffness matrix. However, this can lead to complex formulations of the material tangent moduli that can be difficult to implement or may require custom FE codes, thus limiting their general use. Here we present an approach, based on work by Miehe (Miehe, 1996, “Numerical Computation of Algorithmic (Consistent) Tangent Moduli in Large Strain Computational Inelasticity,” Comput. Methods Appl. Mech. Eng., 134, pp. 223–240), for an efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE codes. By perturbing the deformation gradient, the material tangent moduli from the Jaumann rate of the Kirchhoff stress are accurately approximated by a forward difference of the associated Kirchhoff stresses. The merit of this approach is that it produces a concise mathematical formulation that is not dependent on any particular material model. Consequently, once the approximation method is coded in a subroutine, it can be used for other hyperelastic material models with no modification. The implementation and accuracy of this approach is first demonstrated with a simple neo-Hookean material. Subsequently, a fiber-reinforced structural model is applied to analyze the pressure-diameter curve during blood vessel inflation. Implementation of this approach will facilitate the incorporation of novel hyperelastic material models for a soft tissue behavior into commercial FE software.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

With the material constants G=44.24kPa, k1=0.206kPa, k2=1.465, and D=2.0×10−4kPa−1 and fiber orientations of ±39.76deg, pressure-radius results from the FE model were compared with experimental data, adapted from Fig. 7 of Zulliger (16)

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Figure 2

The FE model for a rat carotid artery segment, (A) before deformation, (B) deformed geometry after a 25kPa inner pressure in a maximum principal stress contour plot, (C) a radial segment of the FE model showing the stress distribution across the artery wall thickness. The FE model geometry, material constants, and loading conditions were adopted from Zulliger (16).

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Figure 3

Maximum principal stress across the artery wall thickness under different static pressures

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Figure 4

Pressure-radius plots of simulated artery inflations with the variation in incompressibility control parameter D with values of 2.0×10−2, 2.0×10−3, 2.0×10−4, and 2.0×10−5



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